It is interesting to write a summary for this collection of works by Kierkegaard mostly because it is completely non-traditional in one aspect: all of the works are chopped into important "selections"! In other words, no work is included in its entirety! Therefore, instead of giving a huge summary of every single selection, let me give you a good overview of what you will find here.
The works included in A Kierkegaard Anthology span the entirety of Kierkegaard's career as a writer. Despite my angst about chopping up Kierkegaard's works, I must say that each selection can be considered episodic in itself in that it does have its own value. Kierkegaard can be considered a "cynical Christian" in that he always remained steadfast in both ideologies (despite how much of an irony that is). A philosopher at heart, Kierkegaard almost always deals in abstractions and, therefore, is particularly hard to read at times. Even though Existentialism didn't really exist yet for Kierkegaard, he is often lumped into that category due to his cynicism and abstraction and, particularly, due to his focus on despair and alienation. In the works of A Kierkegaard Anthology there are three main themes: aestheticism, ethics, and religion.
It is appropriate here to comment on some of the more famous of the selections in A Kierkegaard Anthology. If you look at his Journal, you will find quite a selection of what can be called "platitudes" and epigrams. Contrary to this particular style, we have "Diary of the Seducer" and the "Banquet" scene, that show Kierkegaard to be both a dramatist and a lyricist. However, my favorite is his commentary on religion that can be found in Attack upon "Christendom" where Kierkegaard masters the art of satire.
There are many works in A Kierkegaard Anthology. Although it would be daunting to summarize every one (and perhaps even more difficult to provide a summary for the short and/or long "clips" that are included in the anthology) a full list is quite appropriate here. A comprehensive list is as follows: The Journals, Either/Or, Two Edifying Discourses, Fear and Trembling, Repetition, Philosophical Fragments, Stages on Life's Way, Concluding Unscientific Postscript, The Present Age, Edifying Discourses in Various Spirits, Works of Love, The Point of View for My Work as an Author, The Sickness Unto Death, Training in Christianity, Two Discourses at the Communion on Fridays, The Attack Upon Christendom, and The Unchangeableness of God.
In conclusion, let me mention that this book was originally published quite a long time ago and, therefore, the works by this author are experiencing an interesting revival at this time. One can see why. A Kierkegaard Anthology proves Kierkegaard to be quite a distinguished writer and philosopher.